Analysis on fractals

Analysis on fractals or calculus on fractals is a generalization of calculus on smooth manifolds to calculus on fractals. Fractal calculus or calculus on fractal was formulated in a seminal paper by Parvate and Gangal based on ordinary calculus which is called F<\alpha>-Calculus. Differential equations on fractal sets and curves were defined.

The theory describes dynamical phenomena that occur on objects modelled by fractals. It studies questions such as "how does heat diffuse in a fractal?" and "How does a fractal vibrate?"

In the smooth case the operator that occurs most often in the equations modelling these questions is the Laplacian, so the starting point for the theory of analysis on fractals is to define a Laplacian on fractals. This turns out not to be a full differential operator in the usual sense but has many of the desired properties. There are a number of approaches to defining the Laplacian: probabilistic, analytical, or measure-theoretic. Random processes and random variables and processes on totally disconnected fractal sets were defined. Integrals and derivatives of functions on Cantor tartan spaces were defined. The non-local derivatives on fractal Cantor sets were defined. The scaling properties were given for both local and non-local fractal derivatives. The local and non-local fractal differential equations are solved and compared. Related physical models are also suggested. The generalized Laplace and Sumudu transforms involve functions with totally disconnected fractal sets in the real line. Linear differential equations on Cantor-like sets are solved utilizing fractal Sumudu transforms. Random motion of a particle on a fractal curve, using Langevin approach is derived. This involves defining a new velocity in terms of the mass of the fractal curve, as defined in recent work. The geometry of the fractal curve plays an important role in this analysis. A Langevin equation with a particular model of noise was proposed and solved using techniques of the Fractal Calculus. A new calculus on fractal curves, such as the von Koch curve, was formulated. A Fokker–Planck equation on fractal curves was obtained, starting from Chapmann–Kolmogorov equation on fractal curves.

See also

References
  • Christoph Bandt; Siegfried Graf; Martina Zähle (2000). Fractal Geometry and Stochastics II. Birkhäuser. ISBN 978-3-7643-6215-7.
  • Jun Kigami (2001). Analysis on Fractals. Cambridge University Press. ISBN 978-0-521-79321-6.
  • Robert S. Strichartz (2006). Differential Equations on Fractals. Princeton. ISBN 978-0-691-12542-8.
  • Pavel Exner; Jonathan P. Keating; Peter Kuchment; Toshikazu Sunada & Alexander Teplyaev (2008). Analysis on graphs and its applications: Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January 8-June 29, 2007. AMS Bookstore. ISBN 978-0-8218-4471-7.


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